Mathematics for the interested outsider

AMS Sectional, Day 1

I don’t have wireless access here in the lecture hall, so I can’t “live blog”. I’m writing notes on the lectures I find noteworthy.

David Radford spoke about something called the Hennings Invariant of a finite-dimensional Hopf algebra. I’ve always liked his style, since he manages to boil down a lot of complicated algebraic structure to what’s essential for the application at hand. He also describes it incredibly clearly. His lectures are very accessible to a grad student who has a basic background in algebra, which is more than I can say about many algebraists. I think it should be clear why I think this is a Good Thing.

I’ll get to Hopf algebras in more depth eventually, but for now let me say this: they’re very much like groups, but using somewhat heavier machinery. In the long run, groups and Hopf algebras both work off of a very similar structure.

Pat Gilmer gave a talk on “congruence” and “similarity” of 3-manifolds. A 3-manifold is a space that looks close-up like three-dimensional space, like the surface of the Earth looks flat since we’re so close to it. These two concepts he’s pushing are equivalence relations. Two 3-manifolds may be different, but might still be “similar” or “congruent” if they’re related by certain modifications, called “surgeries”. Congruence was evidently studied about ten years ago and Gilmer reinvented it himself along with similarity. He’s particularly interested in how certain well-known invariants of 3-manifolds change as you apply these surgeries.

One interesting thing this brings to mind is the fact that we can get any 3-manifold from the 3-sphere (the surface of the Earth is a *2*-sphere) by cutting out a bunch of bagel-shaped regions that might be knotted, twisting up the boundaries of the parts we cut out, and putting them back in. This means that there’s a connection between knot theory and 3-manifold theory. In fact, a very large portion of mathematicians calling themselves knot theorists are really more interested in 3-manifolds and just use knot theory as a tool.

After lunch, Carmen Caprau gave her talk about an tangle homology. It manages to fix a big problem I’ve had with Khovanov’s homology theory — it tends to screw up the signs. Knot homology theories are really big business these days. Most people I know who are on the job market and work directly with these sorts of things have jobs nailed down, and Carmen is no exception. Good luck to her.

Scott Carter talked about cohomology in symmetric monoidal categories with products and coproducts. This extends the stuff he has done with Alissa Crans, Mohammed Elhamdadi, Pedro Lopes, and Masahico Saito. The last version of this talk I saw at last Spring’s Knots In Washington was some of the nicest theory I’ve seen. Now they’re taking this abstract setup describing Hoschschild homology theories (which try to capture the underlying essence of associativity), and “dualize” all the diagrams to get some sort of topological invariant. I always love mixing up notation and subject matter, and this is very much in that Kauffman-esque spirit. Hopefully there will be an updated version of their paper on the arXiv soon.

Maciej Niebrzydowski spoke on homology of dihedral quandles, which he worked on with his advisor, Jozef Przytycki. I’ll leave this alone since I’m almost ready to talk about quandles in full detail.

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.